## Variance reduction approaches in stochastic homogenization

#### Frederic LegollEcole Nationale Des Ponts et Chaussees (LAMI)

The simulation of random heterogeneous materials is often very expensive.For instance, in a homogenization setting, the homogenized matrix is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where the corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function is a challenging computational task.

In practice, the corrector problem is solved on a truncated domain, and the exact homogenized matrix is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized matrix turns out to be stochastic, whereas the exact homogenized matrix is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the(approximated) homogenized matrix within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost.

In this talk, we present several variance reduction approaches to address this question, some of them being based on a surrogate, defect-type model proposed by A. Anantharaman and C. Le Bris.

Joint work with X. Blanc, R. Costaouec, C. Le Bris and W. Minvielle.

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